In several parts of their target article, Cohen Kadosh & Walsh (CK&W) highlight the importance of taking a developmental perspective in explorations of the brain's representation of number. However, their article does not provide a model of how non-abstract representation of number in the parietal cortex might arise over developmental time. Since many of the representational formats described by CK&W, such as Arabic numerals and number words, are cultural inventions, their brain representation(s) must be the outcome of a developmental process. In other words, the theory raises the question of why development would involve the construction of a system with multiple format-specific representations of numerical magnitude in the parietal cortex?
In this commentary, I explore the notion of non-abstract representations of numerical magnitude from a developmental perspective and contend that currently available theory and evidence suggests that abstract representations of numerical magnitude are a more plausible outcome of development than non-abstract representations.
According to several current theoretical proposals (Dehaene Reference Dehaene1997; Verguts & Fias Reference Verguts and Fias2004), the acquisition of exact, symbolic representations of number requires the interaction between, on the one hand, preverbal systems that have a long evolutionary history and are shared between species and, on the other hand, language-related, symbolic representation of number that are the product of cultural history. Specifically, evidence suggests that infants and nonverbal animals share the ability to discriminate between large, non-symbolic numerosities (such as arrays of dots), and that infants discrimination abilities are, consistent with Weber's law, ratio dependent (for a review, see Brannon Reference Brannon2006). In addition to this approximate system for the representation of large numbers, infants are thought to have a system for the precise representation of small sets of objects (Feigenson et al. Reference Feigenson, Dehaene and Spelke2004), which is thought to support the ability of children and adults to rapidly (without counting) enumerate small (1–4) items (frequently called “subitizing”). These early representational systems are thought to play an important role in children's acquisition of higher-level numerical skills. In particular, recent evidence suggests that infants' representation of small sets scaffolds their understanding of the meaning of counting (Le Corre & Carey Reference Le Corre and Carey2007). Subsequent to children's gaining understanding of the meaning of number words, numerical symbols such as Arabic numerals are learnt and presumably these initially meaningless symbols acquire their meaning by being mapped onto corresponding number words, which are in turn mapped onto preverbal systems for the representation of numerical magnitude. Thus, development is thought to involve the progressive acquisition of the mappings between different external representations of number. Thus, while the processes that are involved in mapping from external to internal representations may differ between stimulus formats, the internal semantic referent does not differ between representation formats. It is this common representation that allows for the translation between formats. This is also the prediction made by Verguts and Fias, who argue that symbolic and non-symbolic representations of numerical magnitude are mapped onto a shared representation of numerical magnitude subserved by the intraparietal cortex via different pathways. Recent empirical work has supported this proposal by showing format-general representation of quantity in the intraparietal sulcus, as well as format-specific activation in other brain regions (Ansari & Holloway Reference Ansari and Holloway2008; Santes et al., in press). In other words, format-specificity lies in the process of mapping between, on the one hand, different external representations (i.e., number words to Arabic numerals) and, on the other hand, the mapping between external representation and a common, format-general, internal representation of numerical magnitude. This theory is different from the model put forward by CK&W in which the representation of numerical magnitude itself is predicted to be format-specific.
Given that development is thought to involve the progressive interconnection between different external representations that refer to a common internal representation of numerical magnitude, it seems more plausible that development involves the progressive specialization of the parietal cortex for format-independent rather than format-dependent representations of numerical magnitude, while other brain regions might mediate between representations and subserve the association between each external representation and the common, abstract representation of numerical magnitude.
Indeed, a recent neuroimaging study (Cantlon et al., in press) provides evidence to suggest that both children and adults exhibit common activation of the inferior parietal cortex during the processing of symbolic and non-symbolic numerical magnitude. However, children additionally activate prefrontal regions, which may mediate the association between different formats. Such data are consistent with the interactive specialization model of functional brain development, in which functional brain specialization is the product of the interaction between multiple brain regions (Johnson Reference Johnson2001).
If the proposal by CK&W is indeed correct, then the current models of the development of numerical magnitude representations need to be radically revised. If the semantic representation, for example, of number symbols differs qualitatively from the representation of number words and non-symbolic representations of numerical magnitude, then the development of children's understanding of these external representations must involve independent developmental trajectories. Specifically, different external representations of numerical magnitude would be expected to acquire their meaning independently of one another, rather than through becoming interconnected. Furthermore, this developmental trajectory would differ between speakers of different languages and second-language acquisition would be predicted to involve the construction of a new parietal representation for the number words in the newly acquired language. What are the neurocognitive processes that allow for the construction of these format-specific semantic representations of numerical magnitude? The implications of such independent representations might be that children cannot use their semantic representation of number words in order to understand the meaning of Arabic numerals. Furthermore, this would have educational repercussions and may lead to less focus on the relationships between different formats of representations in the classroom.
Taken together, in its current form, the proposal for non-abstract representations put forward by CK&W does not account for the emergence of non-abstract representations over the course of learning and development. Because different formats for the external representation of numerical magnitude are acquired over the course of learning and development, the non-abstract theory must be put to the developmental test.
